Complex Methods | Part IB, 2004

Let the functions ff and gg be analytic in an open, nonempty domain Ω\Omega and assume that g0g \neq 0 there. Prove that if f(z)g(z)|f(z)| \equiv|g(z)| in Ω\Omega then there exists αR\alpha \in \mathbb{R} such that f(z)eiαg(z)f(z) \equiv e^{i \alpha} g(z).

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